Critical exponents of fluid-fluid interfacial tensions near a critical endpoint in a nonwetting gap

TL;DR

The paper investigates interfacial tensions in fluid three-phase equilibria near lines of critical endpoints that meet at a tricritical point, using a mean-field two-density density-functional theory with Griffiths scaling. It shows that the critical interfacial tension vanishes with the mean-field exponent $3/2$ in both wet and nonwet regimes near CEPs, but reveals an exact anomalous exponent $3/4$ at the neutral point where $\sigma_{\alpha\beta}=\sigma_{\alpha\gamma}$ (t=1/2), due to concurrent vanishing of two geometric contributions from the three-phase and Neumann triangles. The anomaly is derived analytically from an $\epsilon$-expansion and explained via geometric arguments, providing a precise prediction for a special path to CEPs. The findings motivate renewed experimental tests of nonwetting gaps and wetting transitions near CEPs, with potential observations of the $3/4$ scaling in suitable oil–water–surfactant mixtures.

Abstract

Fluid three-phase equilibria, with phases $α, β, γ$, are studied close to a tricritical point, analytically and numerically, in a mean-field density-functional theory with two densities. Employing Griffiths' scaling for the densities, the interfacial tensions of the wet and nonwet interfaces are analysed. The mean-field critical exponent is obtained for the vanishing of the critical interfacial tension $σ_{βγ}$ as a function of the deviation of the noncritical interfacial tension $σ_{αγ}$ from its limiting value at a critical endpoint $σ_{α,βγ}$. In the wet regime, this exponent is $3/2$ as expected. In the nonwetting gap of the model, the exponent is again $3/2$, except for the approach to the critical endpoint on the neutral line where $σ_{αβ} = σ_{αγ}$. When this point is approached along any path with $σ_{αβ} \neq σ_{αγ}$, or along the neutral line, $σ_{βγ} \propto | σ_{αγ} - σ_{α,βγ}|^{3/4}$, featuring an anomalous critical exponent $3/4$, which is an exact result derived by analytic calculation and explained by geometrical arguments.

Figures (4)

• Figure 1: Interfacial tensions $\sigma_{\alpha\beta}$ and $\sigma_{\beta\gamma}$ of the interfaces $\alpha\beta$ and $\beta\gamma$, respectively, as well as the interfacial tension $\sigma_{\alpha\gamma}$ of the $\alpha\gamma$ interface that is wet by $\beta$, versus $\sigma_{\alpha\beta}$ for a sweep (varying $s$) between the two CEPs at $t = 0.1$. In this regime close to the TCP, only wet states are encountered and CPW holds. Note that the plot of $\sigma_{\beta\gamma}$ is convex towards the origin as was found experimentally in a specific fluid mixture Langetal. The three-fluid interface configurations asymptotically close to the CEPs, on the edge of three-phase coexistence, are drawn in the diagrams. In these diagrams, the solid line represents the non-critical interface and the dashed line the diffuse near-critical interface.
• Figure 2: Interfacial tensions $\sigma_{\alpha\beta}$, $\sigma_{\beta\gamma}$ and $\sigma_{\alpha\gamma}$ of the interfaces $\alpha\beta$, $\beta\gamma$ and $\alpha\gamma$, respectively, versus $\sigma_{\alpha\beta}$ for a sweep (varying $s$) between the two CEPs at $t = 0.25$. Note that since $\sigma_{\alpha\gamma} < \sigma_{\alpha\beta} +\sigma_{\beta\gamma}$ only nonwetting states are encountered and CPW is absent. The CEPs lie in the nonwetting gap. Note that the plot of $\sigma_{\beta\gamma}$ is not convex towards the origin, and on this scale appears to be almost linear. However, asymptotically close to the CEPs $\sigma_{\beta\gamma}$ approaches the axes tangentially, and features the exponent 3/2 in accord with \ref{['generic']}. The three-fluid interface configurations asymptotically close to the CEPs, on the edge of three-phase coexistence, are drawn in the diagrams. In these diagrams the solid line represents the non-critical interface and the dashed line the diffuse near-critical interface. Note that the asymptotic state is nonwet, with asymptotic contact angle $\hat{\beta} \approx 0.61$ rad in accord with the exact results derived in IK.
• Figure 3: Interfacial tensions $\sigma_{\alpha\beta}$, $\sigma_{\beta\gamma}$ and $\sigma_{\alpha\gamma}$ of the interfaces $\alpha\beta$, $\beta\gamma$ and $\alpha\gamma$, respectively, versus $\sigma_{\alpha\beta}$ for a sweep (varying $s$) between the two CEPs at $t = 1$. Note that since $\sigma_{\alpha\gamma} < \sigma_{\alpha\beta} +\sigma_{\beta\gamma}$ only nonwetting states are encountered and CPW is absent. The CEPs lie in the nonwetting gap. Note that various interfacial tensions have become multi-valued near the $\beta\gamma$ CEP, when plotted as a function of $\sigma_{\alpha\beta}$, while they are single-valued functions of $s$ (not shown here). The point where all three interfacial tensions intersect is the fully symmetric configuration with all three dihedral angles equal to $2\pi/3$. Note that the plot of $\sigma_{\beta\gamma}$ is mostly concave towards the origin, except very close to the CEPs where, once again, $\sigma_{\beta\gamma}$ approaches the axes tangentially (not visible on this scale), and features the exponent 3/2 in accord with \ref{['generic']}. The three-fluid interface configurations asymptotically close to the CEPs, on the edge of three-phase coexistence, are drawn in the diagrams. In these diagrams the solid line represents the non-critical interface and the dashed line the diffuse near-critical interface. Note that the asymptotic state is nonwet, with asymptotic contact angle $\hat{\beta} \approx 2.54$ rad in accord with the exact results derived in IK.
• Figure 4: Interfacial tensions $\sigma_{\alpha\beta}$, $\sigma_{\beta\gamma}$ and $\sigma_{\alpha\gamma}$ of the interfaces $\alpha\beta$, $\beta\gamma$ and $\alpha\gamma$, respectively, versus $\sigma_{\alpha\beta}$ for a sweep (varying $s$) between the two CEPs at $t = 1/2$. Here neutral lines meet the CEPs. Note that the sweep is not along a neutral line. Since $\sigma_{\alpha\gamma} < \sigma_{\alpha\beta} +\sigma_{\beta\gamma}$ only nonwetting states are encountered and CPW is absent. The CEPs lie in the nonwetting gap. Note that the plot of $\sigma_{\beta\gamma}$ is concave towards the origin. Asymptotically close to the CEPs, $\sigma_{\beta\gamma}$ approaches the axes orthogonally, and features the anomalous exponent 3/4 in accord with \ref{['anomaly1']} and \ref{['anomaly2']}. The three-fluid interface configurations asymptotically close to the CEPs, on the edge of three-phase coexistence, are drawn in the diagrams. In these diagrams the solid line represents the non-critical interface and the dashed line the diffuse near-critical interface. Note that the asymptotic state is nonwet, with asymptotic contact angle $\hat{\beta} = \pi/2$.